Maria Bayard Dühring - Solid Mechanics

optimization by approximating the objective function F as function of the frequency using Pade´ expansions,
see Ref. [32]. The room from Fig. 1 is then optimized for the frequency interval [18;23] Hz using five target
frequencies where the target frequencies are updated every 25th iteration step. The quantities used in the
optimization are b ¼ 0:85, hmax ¼ 0:3 m and rmin ¼ 1:5hmax. The optimized design obtained after 415 iterations
is illustrated in Fig. 5 together with the response curve for the initial design and the optimized design. The
objective function is reduced from 111.2 to 75.7 dB. It is seen from the two response curves that the objective
function is minimized in the entire interval for the optimized design. Two of the high peaks have been moved
out of the interval and the last one has been significantly reduced. The solid material in the optimized design is
distributed such that a kind of Helmholtz resonator is formed.
3.2. Optimization of a rectangular room in 3D
ARTICLE IN PRESS
M.B. Dühring et al. / Journal of Sound and Vibration 317 (2008) 557–575 565
Fig. 5. (a) The optimized design for the frequency interval [18;23] Hz and (b) the frequency response for the initial design and the
optimized design.
The optimization problem is now extended to 3D problems and a rectangular room with the geometry
shown in Fig. 6 is considered. We optimize two examples for intervals around the first natural frequency
f ¼ 42:92 Hz for the room with a vertical nodal plane at x ¼ 2 m and high sound pressure amplitude at the end
walls at x ¼ 0 and 4 m. The quantities used are b ¼ 0:5, hmax ¼ 0:4 m and rmin ¼ 0:5hmax and the target
frequencies are updated for each 15 iterations. For the first example the optimization is done for the interval
[41.5;44.5] Hz and one target frequency. After 252 iterations the objective function C is reduced from 115.2 to
81.6 dB. The optimized design and the response for the initial design and the optimized design are seen in
Fig. 7. It is observed that the solid material is distributed at the walls with high sound pressure amplitude and
that the natural frequency has been moved to a higher value outside the interval. For the next example the
frequency interval is extended to [40.5;45.5] Hz and four target frequencies are used. C is minimized from 96.6
to 46.9 dB in 218 iterations and the results are illustrated in Fig. 8. In this case most of the solid material is
distributed at the nodal plane around x ¼ 2 m rather than at the two shorter walls opposite to the previous
example. From the response curve in Fig. 8 it is seen that the first natural frequency is not contained in the
interval after the optimization and instead a natural frequency has appeared at a lower value. These two
examples show that if the frequency interval is slightly changed, the optimization can converge to two very
different, in fact opposite designs. The optimized designs here depend on what side of the interval the first
natural frequency is moved to after the optimization. The best design in this case is the design where the solid
material is placed at low pressure amplitude for the initial design. The two examples here can thus be
interpreted as an extension to 3D of the examples in Fig. 4.